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280 X EXISTENCE THEOREMS

Consider now K^m as the product Ej x E₃ with E₃ = K^m~^p. Let T be the
linear bijection of E₃ onto the supplement Q of P in F generated by
d_p+l,..., d_m, which maps the canonical basis of K^m~^p onto d_p+l9..., d_m. We
define i?(z_1? z₃) =/₁(z₁) + r(z₃) for ^ e F, z₃ 6 I^m~^p; it is obviously (8.9.1) a
continuously differentiate mapping. By definition, we have H(v(z_l9z₃)) =
^(/i(^zi)) ⁼ ^rzi J hence the relation v(z_l,z₃) = v(z{, z₃) implies z{ = Zj, and then
boils down to T(z₃) = r(z₃), which yields z₃ = z₃; therefore v is injective. The
relation S_x = 0 proved above shows that for any z_l e P⁷,//^) = rZ^ where x
is any point in U such that/(jc) =/₁(z₁); the derivative of v at (z_1? z₃) is
therefore the linear mapping (/₁₉ t₃)-*rL_x • t_v + 7(/₃) ((8.9.1) and (8.1.3)).
But as the restriction of H to P_x is injective, P^ is a supplement of Q in F, hence
t/(z_l9 z₃) is a linear homeomorphism of K^m onto F. For any point (z_1} z₃) e F¹,
there is therefore an open neighborhood W of that point in I^m such that the
restriction of v to W is a homeomorphism of W onto an open subset y(W)
of F, by (10.2.5). As in addition v is injective, it is a homeomorphism of
I^m onto the open subset V = v(l^m), whose inverse is continuously differentiable
in V. The relation/= v °/₀ ° u then follows from the definitions.

(10.3.2) If the rank of f(d) is equal to n (resp. to m\ then the conclusion of
(10.3.1) holds with p = n (resp. p = m).

Indeed, at the beginning of this section we have seen that there exists then
a neighborhood of a in which the rank of/'(X) is ^n (resp. ^zm), hence equal
to n (resp. to ni) since it is always at most equal to inf(/w, n) (A.4.18).

PROBLEMS

1. Let E, F be two Banach spaces, A an open neighborhood of a point x₀ e E,/a con-
tinuously differentiable mapping of A into F.

(a) Suppose /'(-^o) is a linear homeomorphism of E onto its image in F; show that
there exists a neighborhood U ^c A of X_Q such that /is a homeomorphism of U onto
/(U) (use Problem 3 of Section 10.1).

(b) Suppose f'(x₀) is surjective; then there exists a number a > 0 having the following
property: if Nisthekernelof/'(;co),for every se E, one has \\f'(xo) -s\\ ^ a-mf\\t+s\\

tsN

(12.16.12). Show that there exists a neighborhood V <= A of x₀ such that/(V) is a neigh-
borhood of/(*₀) in F (use Problem 8 of Section 10.1).

2. Let A be an open subset of C^p, and /an analytic mapping of A into C". Show that if
/is infective, then the rank of D/(x) is equal to^ for every jc e A. (Use contradiction, and
induction on p\ for p— 1, apply Rouche's theorem (9.17.3). Assume Df(a) has a
rank <p for some a e A; show first that after performing a linear transformation in F,
one may assume that, if /(z) = (/j(z), ...,/_p(z)), then DJL/^^O, and if g(z)=*
(/2(^X •-. ,f_p(z)), the rank of Dg(a) is exactly p~ 1; then there is a neighborhood



	280 X EXISTENCE THEOREMS Consider now K^m as the product Ej x E₃ with E₃ = K^m~^p. Let T be the linear bijection of E₃ onto the supplement Q of P in F generated by d_p+l,..., d_m, which maps the canonical basis of K^m~^p onto d_p+l9..., d_m. We define i?(z_1? z₃) =/₁(z₁) + r(z₃) for ^ e F, z₃ 6 I^m~^p; it is obviously (8.9.1) a continuously differentiate mapping. By definition, we have H(v(z_l9z₃)) = ^(/i(^zi)) ⁼ ^rzi J hence the relation v(z_l,z₃) = v(z{, z₃) implies z{ = Zj, and then boils down to T(z₃) = r(z₃), which yields z₃ = z₃; therefore v is injective. The relation S_x = 0 proved above shows that for any z_l e P⁷,//^) = rZ^ where x is any point in U such that/(jc) =/₁(z₁); the derivative of v at (z_1? z₃) is therefore the linear mapping (/₁₉ t₃)-rL_x • t_v* + 7(/₃) ((8.9.1) and (8.1.3)). But as the restriction of H to P_x is injective, P^ is a supplement of Q in F, hence t/(z_l9 z₃) is a linear homeomorphism of K^m onto F. For any point (z_1} z₃) e F¹, there is therefore an open neighborhood W of that point in I^m such that the restriction of v to W is a homeomorphism of W onto an open subset y(W) of F, by (10.2.5). As in addition v is injective, it is a homeomorphism of I^m onto the open subset V = v(l^m), whose inverse is continuously differentiable in V. The relation/= v °/₀ ° u then follows from the definitions.

	(10.3.2) If the rank of f(d) is equal to n (resp. to m\ then the conclusion of (10.3.1) holds with p = n (resp. p = m). Indeed, at the beginning of this section we have seen that there exists then a neighborhood of a in which the rank of/'(X) is ^n (resp. ^zm), hence equal to n (resp. to ni) since it is always at most equal to inf(/w, n) (A.4.18).

	PROBLEMS 1. Let E, F be two Banach spaces, A an open neighborhood of a point x₀ e E,/a con- tinuously differentiable mapping of A into F. (a) Suppose /'(-^o) is a linear homeomorphism of E onto its image in F; show that there exists a neighborhood U ^c A of X_Q such that /is a homeomorphism of U onto /(U) (use Problem 3 of Section 10.1). (b) Suppose f'(x₀) is surjective; then there exists a number a > 0 having the following property: if Nisthekernelof/'(;co),for every se E, one has \\f'(xo) -s\\ ^ a-mf\\t+s\\ tsN (12.16.12). Show that there exists a neighborhood V <= A of x₀ such that/(V) is a neigh- borhood of/(₀) in F (use Problem 8 of Section 10.1). 2. Let A be an open subset of C^p, and /an analytic mapping of A into C". Show that if /is infective,* then the rank of D/(x) is equal to^ for every jc e A. (Use contradiction, and induction on p\ for p— 1, apply Rouche's theorem (9.17.3). Assume Df(a) has a rank <p for some a e A; show first that after performing a linear transformation in F, one may assume that, if /(z) = (/j(z), ...,/_p(z)), then DJL/^^O, and if g(z)=* (/2(^X •-. ,f_p(z)), the rank of Dg(a) is exactly p~ 1; then there is a neighborhood